TD-pairs and the 𝑞-Onsager algebra

Ito, Tatsuro

doi:10.1090/suga/444

TD-pairs and the $q$-Onsager algebra
HTML articles powered by AMS MathViewer

by Tatsuro Ito
Translated by: Tatsuro Ito

Sugaku Expositions 32 (2019), 205-232

DOI: https://doi.org/10.1090/suga/444

Published electronically: September 26, 2019

PDF | Request permission

References

Richard Askey and James Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or $6-j$ symbols, SIAM J. Math. Anal. 10 (1979), no. 5, 1008–1016. MR 541097, DOI 10.1137/0510092
Christine Bachoc and Frank Vallentin, New upper bounds for kissing numbers from semidefinite programming, J. Amer. Math. Soc. 21 (2008), no. 3, 909–924. MR 2393433, DOI 10.1090/S0894-0347-07-00589-9
Eiichi Bannai, Algebraic combinatorics—recent topics in association schemes, Sūgaku 45 (1993), no. 1, 55–75 (Japanese). MR 1236220
Eiichi Bannai, Combinatorics regarded as pure mathematics. The aims of algebraic combinatorics, Sūgaku 62 (2010), no. 4, 433–452 (Japanese). MR 2789097
Ei. Bannai, Et. Bannai and T. Ito, Introduction to Algebraic Combinatorics (in Japanese) Kyoritsu Shuppan, 2016, vii + 511 pages.
Eiichi Bannai and Tatsuro Ito, Algebraic combinatorics. I, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984. Association schemes. MR 882540
Pascal Baseilhac, Deformed Dolan-Grady relations in quantum integrable models, Nuclear Phys. B 709 (2005), no. 3, 491–521. MR 2123215, DOI 10.1016/j.nuclphysb.2004.12.016
Pascal Baseilhac, An integrable structure related with tridiagonal algebras, Nuclear Phys. B 705 (2005), no. 3, 605–619. MR 2114110, DOI 10.1016/j.nuclphysb.2004.11.014
Pascal Baseilhac and Kozo Koizumi, A new (in)finite-dimensional algebra for quantum integrable models, Nuclear Phys. B 720 (2005), no. 3, 325–347. MR 2153659, DOI 10.1016/j.nuclphysb.2005.05.021
Georgia Benkart and Paul Terwilliger, Irreducible modules for the quantum affine algebra $U_q(\widehat {sl}_2)$ and its Borel subalgebra, J. Algebra 282 (2004), no. 1, 172–194. MR 2095578, DOI 10.1016/j.jalgebra.2004.08.016
Norman Biggs, Algebraic graph theory, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1993. MR 1271140
A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-regular graphs, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 18, Springer-Verlag, Berlin, 1989. MR 1002568, DOI 10.1007/978-3-642-74341-2
Vyjayanthi Chari and Andrew Pressley, Quantum affine algebras, Comm. Math. Phys. 142 (1991), no. 2, 261–283. MR 1137064
E. R. van Dam and J. H. Koolen, A new family of distance-regular graphs with unbounded diameter, Invent. Math. 162 (2005), no. 1, 189–193. MR 2198328, DOI 10.1007/s00222-005-0442-3
Etsuro Date and Shi-shyr Roan, The structure of quotients of the Onsager algebra by closed ideals, J. Phys. A 33 (2000), no. 16, 3275–3296. MR 1766989, DOI 10.1088/0305-4470/33/16/316
Brian Davies, Onsager’s algebra and superintegrability, J. Phys. A 23 (1990), no. 12, 2245–2261. MR 1063559
Brian Davies, Onsager’s algebra and the Dolan-Grady condition in the non-self-dual case, J. Math. Phys. 32 (1991), no. 11, 2945–2950. MR 1131672, DOI 10.1063/1.529036
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. 10 (1973), vi+97. MR 384310
Garth A. Dickie, Q-polynomial structures for association schemes and distance-regular graphs, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 2693006
L. Dolan and Michael Grady, Conserved charges from self-duality, Phys. Rev. D (3) 25 (1982), no. 6, 1587–1604. MR 649050, DOI 10.1103/PhysRevD.25.1587
Yoshimi Egawa, Association schemes of quadratic forms, J. Combin. Theory Ser. A 38 (1985), no. 1, 1–14. MR 773550, DOI 10.1016/0097-3165(85)90016-0
Dion Gijswijt, Alexander Schrijver, and Hajime Tanaka, New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming, J. Combin. Theory Ser. A 113 (2006), no. 8, 1719–1731. MR 2269550, DOI 10.1016/j.jcta.2006.03.010
Tomoya Hattai and Tatsuro Ito, On a certain subalgebra of $U_q(\widehat {\mathfrak {sl}}_2)$ related to the degenerate $q$-Onsager algebra, SIGMA Symmetry Integrability Geom. Methods Appl. 11 (2015), Paper 007, 13. MR 3313683, DOI 10.3842/SIGMA.2015.007
Akihito Hora and Nobuaki Obata, Asymptotic spectral analysis of growing regular graphs, Trans. Amer. Math. Soc. 360 (2008), no. 2, 899–923. MR 2346476, DOI 10.1090/S0002-9947-07-04232-8
T. Ito, The classification of TD-pairs, RIMS Kokyuroku, 1926 (2014), 146-164.
Tatsuro Ito, Kazumasa Nomura, and Paul Terwilliger, A classification of sharp tridiagonal pairs, Linear Algebra Appl. 435 (2011), no. 8, 1857–1884. MR 2810633, DOI 10.1016/j.laa.2011.03.032
Tatsuro Ito, Kenichiro Tanabe, and Paul Terwilliger, Some algebra related to $P$- and $Q$-polynomial association schemes, Codes and association schemes (Piscataway, NJ, 1999) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 56, Amer. Math. Soc., Providence, RI, 2001, pp. 167–192. MR 1816397, DOI 10.1090/dimacs/056/14
Tatsuro Ito and Paul Terwilliger, The shape of a tridiagonal pair, J. Pure Appl. Algebra 188 (2004), no. 1-3, 145–160. MR 2030811, DOI 10.1016/j.jpaa.2003.10.002
Tatsuro Ito and Paul Terwilliger, Tridiagonal pairs and the quantum affine algebra $U_q(\widehat {\mathrm {sl}}_2)$, Ramanujan J. 13 (2007), no. 1-3, 39–62. MR 2281156, DOI 10.1007/s11139-006-0242-4
Tatsuro Ito and Paul Terwilliger, Two non-nilpotent linear transformations that satisfy the cubic $q$-Serre relations, J. Algebra Appl. 6 (2007), no. 3, 477–503. MR 2337765, DOI 10.1142/S021949880700234X
Tatsuro Ito and Paul Terwilliger, The $q$-tetrahedron algebra and its finite dimensional irreducible modules, Comm. Algebra 35 (2007), no. 11, 3415–3439. MR 2362663, DOI 10.1080/00927870701509180
T. Ito and P. Terwilliger, The Drinfel’d polynomial of a tridiagonal pair, J. Combin. Inform. System Sci., 34 (2009), 255–292.
T. Ito and P. Terwilliger, The q-Onsager algebra, In: Finite Groups, Vertex Operator Algebras and Combinatorics (ed. H. Yamada), RIMS Kokyuroku 1656 (2009), 84–88.
Tatsuro Ito and Paul Terwilliger, The augmented tridiagonal algebra, Kyushu J. Math. 64 (2010), no. 1, 81–144. MR 2662661, DOI 10.2206/kyushujm.64.81
Douglas A. Leonard, Orthogonal polynomials, duality and association schemes, SIAM J. Math. Anal. 13 (1982), no. 4, 656–663. MR 661597, DOI 10.1137/0513044
Oleg R. Musin, The kissing number in four dimensions, Ann. of Math. (2) 168 (2008), no. 1, 1–32. MR 2415397, DOI 10.4007/annals.2008.168.1
Kazumasa Nomura, Tridiagonal pairs and the Askey-Wilson relations, Linear Algebra Appl. 397 (2005), 99–106. MR 2116451, DOI 10.1016/j.laa.2004.10.004
Kazumasa Nomura and Paul Terwilliger, The structure of a tridiagonal pair, Linear Algebra Appl. 429 (2008), no. 7, 1647–1662. MR 2444350, DOI 10.1016/j.laa.2008.04.042
Masatoshi Noumi and Katsuhisa Mimachi, Askey-Wilson polynomials as spherical functions on $\textrm {SU}_q(2)$, Quantum groups (Leningrad, 1990) Lecture Notes in Math., vol. 1510, Springer, Berlin, 1992, pp. 98–103. MR 1183481, DOI 10.1007/BFb0101182
N. Obata, Quantum probability and spectral analysis of graphs, In: Proceedings of the 26th Symposium on Algebraic Combinatorics (Japanese), (eds. K. Waki, F. Oda, M. Harada), 2009, 102-125.
Lars Onsager, Crystal statistics. I. A two-dimensional model with an order-disorder transition, Phys. Rev. (2) 65 (1944), 117–149. MR 10315
S.S. Roan, Onsager’s algebra, loop algebra and chiral Potts model, MPI 91-70, Max-Plank-Institut fur Mathematik, Bonn, 1991.
Alexander Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (2005), no. 8, 2859–2866. MR 2236252, DOI 10.1109/TIT.2005.851748
Paul Terwilliger, $P$ and $Q$ polynomial schemes with $q=-1$, J. Combin. Theory Ser. B 42 (1987), no. 1, 64–67. MR 872408, DOI 10.1016/0095-8956(87)90063-3
Paul Terwilliger, Balanced sets and $Q$-polynomial association schemes, Graphs Combin. 4 (1988), no. 1, 87–94. MR 922163, DOI 10.1007/BF01864156
Paul Terwilliger, The subconstituent algebra of an association scheme. I, J. Algebraic Combin. 1 (1992), no. 4, 363–388. MR 1203683, DOI 10.1023/A:1022494701663
Paul Terwilliger, The subconstituent algebra of an association scheme. II, J. Algebraic Combin. 2 (1993), no. 1, 73–103. MR 1210403, DOI 10.1023/A:1022480715311
Paul Terwilliger, The subconstituent algebra of an association scheme. III, J. Algebraic Combin. 2 (1993), no. 2, 177–210. MR 1229433, DOI 10.1023/A:1022415825656
Paul Terwilliger, Two relations that generalize the $q$-Serre relations and the Dolan-Grady relations, Physics and combinatorics 1999 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, pp. 377–398. MR 1865045, DOI 10.1142/9789812810199_{0}013
Paul Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), no. 1-3, 149–203. MR 1826654, DOI 10.1016/S0024-3795(01)00242-7
Paul Terwilliger, Leonard pairs and the $q$-Racah polynomials, Linear Algebra Appl. 387 (2004), 235–276. MR 2069278, DOI 10.1016/j.laa.2004.02.014
Paul Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array, Des. Codes Cryptogr. 34 (2005), no. 2-3, 307–332. MR 2128338, DOI 10.1007/s10623-004-4862-7
Paul Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other: comments on the split decomposition, J. Comput. Appl. Math. 178 (2005), no. 1-2, 437–452. MR 2127896, DOI 10.1016/j.cam.2004.04.017
Paul Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form, J. Algebra 291 (2005), no. 1, 1–45. MR 2158508, DOI 10.1016/j.jalgebra.2005.05.033
Paul Terwilliger and Raimundas Vidunas, Leonard pairs and the Askey-Wilson relations, J. Algebra Appl. 3 (2004), no. 4, 411–426. MR 2114417, DOI 10.1142/S0219498804000940
Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov, From $sl_q(2)$ to a parabosonic Hopf algebra, SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011), Paper 093, 13. MR 2861183, DOI 10.3842/SIGMA.2011.093
Satoshi Tsujimoto, Luc Vinet, and Alexei Zhedanov, Dunkl shift operators and Bannai-Ito polynomials, Adv. Math. 229 (2012), no. 4, 2123–2158. MR 2880217, DOI 10.1016/j.aim.2011.12.020
Luc Vinet and Alexei Zhedanov, A limit $q=-1$ for the big $q$-Jacobi polynomials, Trans. Amer. Math. Soc. 364 (2012), no. 10, 5491–5507. MR 2931336, DOI 10.1090/S0002-9947-2012-05539-5
Luc Vinet and Alexei Zhedanov, A ‘missing’ family of classical orthogonal polynomials, J. Phys. A 44 (2011), no. 8, 085201, 16. MR 2770369, DOI 10.1088/1751-8113/44/8/085201